Abstract：By constructing an appropriate payoff function, quasi-variational inequality, generalized variational inequality, Von Neumann Lemma and the extension of Gale-Nikaido-Debreu Lemma, can be derived from Nash equilibrium theorem, with the aid of the inverse of the Berge maximum theorem. Moreover, an important method is provided that an upper semicontinuous and convex compact set-valued mapping problem is converted into a binary function. Our results and proofs are all new.